Definition:Bounded Below Mapping/Real-Valued

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This page is about Bounded Below in the context of Real-Valued Function. For other uses, see Bounded Below.

Definition

Let $f: S \to \R$ be a real-valued function.


Then $f$ is bounded below on $S$ by the lower bound $L$ if and only if:

$\forall x \in S: L \le \map f x$


That is, if and only if the set $\set {\map f x: x \in S}$ is bounded below in $\R$ by $L$.


Unbounded Below

Let $f: S \to \R$ be a real-valued function.


Then $f$ is unbounded below on $S$ if and only if it is not bounded below on $S$:

$\neg \exists L \in \R: \forall x \in S: L \le \map f x$


Also see


Sources